I want a local client that pushes my working copy to GitLab and runs tests on it.

GitLab can receive the code, commit it to a 'hidden' repo, and run the tests. Sending feedback about failing test or successful tests back to the client.

The client could be part of an official command line client, currently we have community ones under CLI clients on https://about.gitlab.com/applications/ Or we can just make a clean API and contribute support to the most popular community command line clients, avoiding a maintenance burden and allowing people to add this functionality to editors.

If we also have an IDE #12759 (closed) I could see some potential to integrate it closely. But I recommend keeping it simple so that it also works for users not using the IDE.